Identification of a coherent twin relationship from high-resolution reciprocal-space maps

The theory and algorithm are presented for the assignment of ferroelastic domains to the individual components of split Bragg peaks in high-resolution reciprocal-space maps. The formalism of mechanical compatibility of ferroelastic domains is further developed for the analysis of the geometry of the reciprocal space. The application of the algorithm to the reciprocal-space maps of tetragonal BaTiO3 and rhombohedral PbZr1−x Ti x O3 crystals is demonstrated.


Figure S1
The same as Figure 6 but for the case of intensity distribution around 002 reflection from multi-domain BaTiO3 crystal.

Table S1
The summary of the individually marked sub-peaks in the 002 reciprocal space map.
The top row gives the peaks numbers (per their marking in Figure S1), the second row gives the corresponding length of the reciprocal space vector (| | = + + ), the third row gives the best matching calculated length of the reciprocal lattice vector (| | = * ), the forth row gives the domain number(s) for which this matching is achieved. Three bottom rows give the reciprocal lattice coordinates (Δ = Δ * + Δ * + Δ * ) of all the peaks relative to the peak center of gravity  Figure 3). The third and four's column indicate the expected separation between the peaks in the analytical and numerical form correspondingly. The fifth column shows the best matching (when reasonable match is found) separation between the Bragg peaks and their numbers according to the Table S1.
Sub peaks pair (According the numbering in Figure S1) Accordingly, the following assignment can be made.
o The sub-peaks 3 and 2 are assigned to the tetragonal domains 1( ) and 3( ). These domains meet along (101 ) oriented domain wall.
o The sub-peaks 5 and 1 are assigned to the tetragonal domains 1( ) and 3( ). These domains meet along (101) oriented domain wall.
Note, that no unambiguous assignment could be done for the sub-peak 4. If such assignment were made based on the length of the reciprocal lattice vector alone then this sub-peak would be associated with either domain 1( ) or domain 2( ).

Figure S2
The same as Figure S1 but for the case of 222 reflection from multi-domain BaTiO3 crystal.

Table S4
The same as Table S2 but for the case of 222 reflection from BaTiO3 crystal.
Sub peaks pair, and (According the numbering in Figure S2) Accordingly, the following assignment can be made: o The sub-peaks 1 and 2 are assigned to the tetragonal domains 1( ) and 2( ). These domains are matched along (11 0) oriented domain wall.
o The sub-peaks 1 and 3 are assigned to the tetragonal domains 1( ) and 3( ). These domains meet along (101 ) domain wall.
o The sub-peaks 5 and 6 can be assigned to the tetragonal domains 1( ) and 3( ). These domains meet along (101 ) domain wall.
o The sub-peaks 2 and 3 are assigned to the tetragonal domains 2( ) and 3( ). These domains meet along (011 ) domain wall.

Figure S3
The same as in the Figure S1 but for the case of 013 reflection from multi-domain BaTiO3 crystal.  Accordingly, the following assignment can be made: o The sub-peaks 3 and 1 are assigned to the tetragonal domains 1( ) and 2( ). These domains meet along (110) domain wall.
o The sub-peaks 2 and 4 can be assigned to the tetragonal domains 1( ) and 3( ) and these peaks are matched along (101 )-oriented domain wall.
o The sub-peaks 3 and 5 can be assigned to the tetragonal domains 1( ) and 3( ) and these peaks are matched along (101) domain wall.  Table S8, S10, S12 (identical to Table   6) illustrate the assignment of the peaks / peaks pairs to the matched domain pairs. These connections are illustrated in Figure S8, S10, S12.  10 Sub peaks pair, and (According the numbering in Figure S4) This example demonstrate the case when ambiguous assignment of sub-peaks to domains is not possible. Any of the two following assignment can be suggested.
o The sub-peaks 2 and 1 are assigned to the rhombohedral domains 2 and 3. These domains meet (001) domain wall.
o Alternatively, the sub-peaks 2 and 1 are assigned to the rhombohedral domains 1 and 4, meeting along (001)-oriented domain wall too.

Figure S5
The same as Figure S1 but for the case of 1 33 reflections from multi-domain PZT crystal.  According to Table S10 and Figure 5 the following assignment can be made: o The sub-peaks 3 and 4 are assigned to the rhombohedral domains 1 and 2. These domains meet along (011) domain wall.
o The sub-peaks 4 and 2 can be assigned to the tetragonal domains 2 and 3. These domains meet along (001) domain wall.

Figure S6
The same as Figure S1 but for the case of 233 reflections from PZT crystal.  According to Table S12 and Figure S6 the following assignment can be made: o The sub-peaks 1 and 2 are assigned to the rhombohedral domains 1 and 2. These domains meet along (011) domain wall.
o The sub-peaks 2 and 3 can be assigned to the tetragonal domains 2 and 3. These domains meet along (001) domain wall.